Shape transitions in a soft incompressible sphere with residual stresses

Residual stresses may appear in elastic bodies, owing to the formation of misfits in the microstructure, driven by plastic deformations or thermal or growth processes. They are especially widespread in living matter, resulting from dynamic remodelling processes aimed at optimizing the overall structural response to environmental physical forces. From a mechanical viewpoint, residual stresses are classically modelled through the introduction of a virtual incompatible state that collects the local relaxed states around each material point. In this work, we employ an alternative approach based on a strain energy function that constitutively depends only on the deformation gradient and the residual stress tensor. In particular, our objective is to study the morphological stability of an incompressible sphere, made of a neo-Hookean material, and subjected to given distributions of residual stresses. The boundary value elastic problem is studied with analytic and numerical tools. Firstly, we perform a linear stability analysis on the prestressed solid sphere using the method of incremental deformations. The marginal stability conditions are given as a function of a control parameter, which is the dimensionless variable that represents the characteristic intensity of the residual stresses. Secondly, we perform finite-element simulations using a mixed formulation in order to investigate the postbuckling morphology in the fully nonlinear regime. Considering different initial distributions of the residual stresses, we find that different morphological transitions occur around the material domain, where the hoop residual stress reaches its maximum compressive value. The loss of spherical symmetry is found to be controlled by the mechanical and geometrical properties of the sphere, as well as the spatial distribution of the residual stress. The results provide useful guidelines for designing morphable soft spheres, for example by controlling residual stresses through active deformations. They finally suggest a viable solution for the nondestructive characterization of residual stresses in soft tissues, such as solid tumours.